DOCSLIB.ORG

Carmichael Numbers and Korselt's Criterion

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Carmichael Numbers and Korselt's Criterion CARMICHAEL NUMBERS AND KORSELT'S CRITERION KEITH CONRAD Fermat's little theorem says for a prime p and all integers a 6≡ 0 mod p that ap−1 ≡ 1 mod p. Conversely, if n > 1 and an−1 ≡ 1 mod n for all a 6≡ 0 mod n then n must be prime: from an−1 ≡ 1 mod n we get (a; n) = 1, so n is relatively prime to all integers from 1 to n − 1, and thus n is a prime number. While a 6≡ 0 mod p is the same as (a; p) = 1 when p is prime and a is any integer, the conditions a 6≡ 0 mod n and (a; n) = 1 are not the same when n is composite: (a; n) = 1 ) a 6≡ 0 mod n but the converse is usually false (depending on a). If we write a 6≡ 0 mod p with (a; p) = 1 in Fermat's little theorem, making it \(a; p) = 1 =) ap−1 ≡ 1 mod p," we have an implication that is sometimes true if we replace prime p by composite n. Definition 1. An integer n > 1 is called a Carmichael number if n is composite and (a; n) = 1 =) an−1 ≡ 1 mod n for all a 2 Z. Initially it is not at all clear that there should be any Carmichael numbers, but the first few were found by Robert Carmichael [1], [2] in the early 20th century and they are 561, 1105, 1729, 2465, 2821. It is possible to verify that an integer n is a Carmichael number without using the definition, so not having to check an−1 ≡ 1 mod n for all a that are relatively prime to n. Instead we can check a property of the prime factorization of n known as Korselt's criterion. Theorem 2 (Korselt). A composite integer n > 2 is a Carmichael number if and only if (i) n is squarefree and (ii) for every prime p dividing n, also (p − 1) j (n − 1). Proof. Assume n is a Carmichael number. First we will show n is squarefree. If a prime p divides n more than once, write n = pkn0 where k ≥ 1 and (p; n0) = 1. We want to show k = 1, and will do this by contradiction using the Chinese remainder theorem. Assume k ≥ 2, so n is divisible by p2. By the Chinese remainder theorem there is an a 2 Z such that a ≡ 1 + p mod pk and a ≡ 1 mod n0. Then (a; n) = 1, so an−1 ≡ 1 mod n by the definition of Carmichael numbers. Reduce the above congruence modulo p2, getting (1 + p)n−1 ≡ 1 mod p2. By the binomial theorem, (1 + p)n−1 ≡ 1 + (n − 1)p mod p2. Since n is divisible by p, 1+(n−1)p ≡ 1−p mod p2. Thus 1−p ≡ 1 mod p2. This is a contradiction, so k = 1. Next we show (p − 1) j (n − 1) for each prime p dividing n. Since n is squarefree, p and n=p are relatively prime. Pick any b 2 Z such that b mod p has order p − 1 (there is a primitive root modulo any prime). By the Chinese remainder theorem there's an a 2 Z such that a ≡ b mod p and a ≡ 1 mod n=p, so (a; n) = 1. Then an−1 ≡ 1 mod n. Reducing both sides modulo p, bn−1 ≡ 1 mod p. This implies, by the choice of b, that (p−1) j (n−1). Now assume n is squarefree and (p − 1) j (n − 1) for every prime p that divides n. We want to show n is Carmichael. If a 2 Z satisfies (a; n) = 1 then for each prime p dividing n we have (a; p) = 1, so ap−1 ≡ 1 mod p. Since p−1 is a factor of n−1 we get an−1 ≡ 1 mod p. 1 2 KEITH CONRAD As this holds for all primes p dividing n, and n is squarefree, we get an−1 ≡ 1 mod n. Also n is composite (hypothesis), so n is Carmichael. Korselt [4] proved this theorem about 10 years before Carmichael essentially rediscovered it [1], [2], but Korselt was unable to find examples of such numbers and that is why they are called Carmichael numbers. On that basis, however, these numbers should be called Simerka˘ numbers since V. Simerka˘ found the first 7 examples 25 years before Carmichael. Below is an excerpt from Simerka's˘ article [5], which appeared in a Czech math journal that was not widely read. Figure 1. Carmichael numbers found by Simerka˘ before Carmichael. Example 3. We use Korselt's criterion to verify a construction of Carmichael numbers due to Chernick [3]: if k is a positive integer such that 6k + 1, 12k + 1, and 18k + 1 are all prime then the product n = (6k + 1)(12k + 1)(18k + 1) is a Carmichael number. For instance, 7 · 13 · 19 = 1729 is a Carmichael number. The first condition of Korselt's criterion, that n be squarefree, obviously holds. To check the second criterion we want to show n − 1 is divisible by 6k, 12k, and 18k. Since 6 j 12 it suffices to look at n mod 12k and n mod 18k. Modulo 12k we have n ≡ (6k + 1)(6k + 1) ≡ 1 mod 12k, and modulo 18k we have n ≡ (6k + 1)(12k + 1) ≡ 1 mod 18k. Thus n − 1 is divisible by the desired factors, so it is a Carmichael number when its three factors are all prime. This method of Chernick is very convenient if you want to construct an example of a large Carmichael number, such as if you are giving a lecture on Carmichael numbers and want to give an example other than the same examples that are in all the books. Run a computer algebra package to find big k where 6k + 1, 12k + 1, and 18k + 1 are all prime (it's believed this should happen infinitely often, and in any case it usually doesn't take long to find such a choice of k), take their product, and you're done. As an exercise, verify with Korselt's criterion that if 6k + 1, 12k + 1, 18k + 1, and 36k + 1 are all prime then their product is a Carmichael number. Corollary 4. A composite integer n is a Carmichael number if and only if an ≡ a mod n for all a 2 Z. Proof. If an ≡ a mod n for all a 2 Z, then when (a; n) = 1 we can cancel a from both sides and get an−1 ≡ 1 mod n, so n is a Carmichael number since it is composite. Conversely, assume n is Carmichael. To prove, for each a 2 Z, that an ≡ a mod n it suffices since n is squarefree to prove an ≡ a mod p for each prime p dividing n. This congruence is obvious if a ≡ 0 mod p. If a 6≡ 0 mod p then ap−1 ≡ 1 mod p by Fermat's n−1 n little theorem, and p−1 is a factor of n−1, so a ≡ 1 mod p, and thus a ≡ a mod p. CARMICHAEL NUMBERS AND KORSELT'S CRITERION 3 Here are some further properties (but not characterizations) of Carmichael numbers. Theorem 5. Every Carmichael number pn is odd, has at least three different prime factors, and every prime factor of n is less than n. Proof. Since n − 1 is relatively prime to n, we have (n − 1)n−1 ≡ 1 mod n, so (−1)n−1 ≡ 1 mod n and we know (−1)n−1 = ±1. Since n > 2 we have −1 6≡ 1 mod n, so (−1)n−1 = 1. Thus n − 1 is even, so n is odd. Suppose n = pq for primes p and q. Since n is squarefree, p 6= q. We may assume without loss of generality that p > q. By Korselt's criterion, (p − 1) j (n − 1). Since n − 1 pq − 1 (p − 1)q + q − 1 q − 1 = = = q + ; p − 1 p − 1 p − 1 p − 1 we must have (p − 1) j (q − 1). But this is impossible since q − 1 < p − 1. Thus n has at least three different prime factors. If p is a prime factor of n, then n − 1 p(n=p) − 1 (p − 1)(n=p) + n=p − 1 n n=p − 1 = = = + ; p − 1 p − 1 p − 1 p p − 1 2 so (p − 1) j (n=p − 1). Thus p ≤ n=p, and the inequality must be strict (otherwise n = p , which is impossible), so p < n. Incidentally,p this givesp another proof thatp pn has at least three prime factors: if n = pq with p < n and q < n then n = pq < n n = n, which is a contradiction. References [1] R. D. Carmichael, Note on a New Number Theory Function, Bulletin Amer. Math. Soc. 16 (1910), 232{238. [2] R. D. Carmichael, On composite P which satisfy the Fermat congruence aP −1 ≡ 1 mod P , Amer. Math. Monthly 19 (1912), 22{27. [3] J. Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc. 45 (1939), 269{274. [4] A. R. Korselt, Probl`emechinois, L'interm´ediairedes math´ematiciens 6 (1899), 142{143. [5]V. Simerka˘ Zbytky z arithemetick´e posloupnosti (On the remainders of an arithmetic progression), Casopis˘ pro p˘estov´an´ı matematiky a fysiky 14 (1885), 221{225. Online at https://gdz.sub. uni-goettingen.de/id/PPN31311028X 0014..
Load more

Recommended publications
  • The Pseudoprimes to 25 • 109
    MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 151 JULY 1980, PAGES 1003-1026 The Pseudoprimes to 25 • 109 By Carl Pomerance, J. L. Selfridge and Samuel S. Wagstaff, Jr. Abstract. The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Car- michael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences. 1. Introduction. According to Fermat's "Little Theorem", if p is prime and (a, p) = 1, then ap~1 = 1 (mod p). This theorem provides a "test" for primality which is very often correct: Given a large odd integer p, choose some a satisfying 1 <a <p - 1 and compute ap~1 (mod p). If ap~1 pi (mod p), then p is certainly composite. If ap~l = 1 (mod p), then p is probably prime. Odd composite numbers n for which (1) a"_1 = l (mod«) are called pseudoprimes to base a (psp(a)). (For simplicity, a can be any positive in- teger in this definition. We could let a be negative with little additional work. In the last 15 years, some authors have used pseudoprime (base a) to mean any number n > 1 satisfying (1), whether composite or prime.) It is well known that for each base a, there are infinitely many pseudoprimes to base a.
    [Show full text]
  • Composite Numbers That Give Valid RSA Key Pairs for Any Coprime P
    information Article Composite Numbers That Give Valid RSA Key Pairs for Any Coprime p Barry Fagin ID Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA; [email protected]; Tel.: +1-719-339-4514 Received: 13 August 2018; Accepted: 25 August 2018; Published: 28 August 2018 Abstract: RSA key pairs are normally generated from two large primes p and q. We consider what happens if they are generated from two integers s and r, where r is prime, but unbeknownst to the user, s is not. Under most circumstances, the correctness of encryption and decryption depends on the choice of the public and private exponents e and d. In some cases, specific (s, r) pairs can be found for which encryption and decryption will be correct for any (e, d) exponent pair. Certain s exist, however, for which encryption and decryption are correct for any odd prime r - s. We give necessary and sufficient conditions for s with this property. Keywords: cryptography; abstract algebra; RSA; computer science education; cryptography education MSC: [2010] 11Axx 11T71 1. Notation and Background Consider the RSA public-key cryptosystem and its operations of encryption and decryption [1]. Let (p, q) be primes, n = p ∗ q, f(n) = (p − 1)(q − 1) denote Euler’s totient function and (e, d) the ∗ Z encryption/decryption exponent pair chosen such that ed ≡ 1. Let n = Un be the group of units f(n) Z mod n, and let a 2 Un. Encryption and decryption operations are given by: (ae)d ≡ (aed) ≡ (a1) ≡ a mod n We consider the case of RSA encryption and decryption where at least one of (p, q) is a composite number s.
    [Show full text]
  • The Distribution of Pseudoprimes
    The Distribution of Pseudoprimes Matt Green September, 2002 The RSA crypto-system relies on the availability of very large prime num- bers. Tests for finding these large primes can be categorized as deterministic and probabilistic. Deterministic tests, such as the Sieve of Erastothenes, of- fer a 100 percent assurance that the number tested and passed as prime is actually prime. Despite their refusal to err, deterministic tests are generally not used to find large primes because they are very slow (with the exception of a recently discovered polynomial time algorithm). Probabilistic tests offer a much quicker way to find large primes with only a negligibal amount of error. I have studied a simple, and comparatively to other probabilistic tests, error prone test based on Fermat’s little theorem. Fermat’s little theorem states that if b is a natural number and n a prime then bn−1 ≡ 1 (mod n). To test a number n for primality we pick any natural number less than n and greater than 1 and first see if it is relatively prime to n. If it is, then we use this number as a base b and see if Fermat’s little theorem holds for n and b. If the theorem doesn’t hold, then we know that n is not prime. If the theorem does hold then we can accept n as prime right now, leaving a large chance that we have made an error, or we can test again with a different base. Chances improve that n is prime with every base that passes but for really big numbers it is cumbersome to test all possible bases.
    [Show full text]
  • On Types of Elliptic Pseudoprimes
    journal of Groups, Complexity, Cryptology Volume 13, Issue 1, 2021, pp. 1:1–1:33 Submitted Jan. 07, 2019 https://gcc.episciences.org/ Published Feb. 09, 2021 ON TYPES OF ELLIPTIC PSEUDOPRIMES LILJANA BABINKOSTOVA, A. HERNANDEZ-ESPIET,´ AND H. Y. KIM Boise State University e-mail address: [email protected] Rutgers University e-mail address: [email protected] University of Wisconsin-Madison e-mail address: [email protected] Abstract. We generalize Silverman's [31] notions of elliptic pseudoprimes and elliptic Carmichael numbers to analogues of Euler-Jacobi and strong pseudoprimes. We inspect the relationships among Euler elliptic Carmichael numbers, strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I, the former two of which we introduce and the latter two of which were introduced by Mazur [21] and Silverman [31] respectively. In particular, we expand upon the work of Babinkostova et al. [3] on the density of certain elliptic Korselt numbers of Type I which are products of anomalous primes, proving a conjecture stated in [3]. 1. Introduction The problem of efficiently distinguishing the prime numbers from the composite numbers has been a fundamental problem for a long time. One of the first primality tests in modern number theory came from Fermat Little Theorem: if p is a prime number and a is an integer not divisible by p, then ap−1 ≡ 1 (mod p). The original notion of a pseudoprime (sometimes called a Fermat pseudoprime) involves counterexamples to the converse of this theorem. A pseudoprime to the base a is a composite number N such aN−1 ≡ 1 mod N.
    [Show full text]
  • Independence of the Miller-Rabin and Lucas Probable Prime Tests
    Independence of the Miller-Rabin and Lucas Probable Prime Tests Alec Leng Mentor: David Corwin March 30, 2017 1 Abstract In the modern age, public-key cryptography has become a vital component for se- cure online communication. To implement these cryptosystems, rapid primality test- ing is necessary in order to generate keys. In particular, probabilistic tests are used for their speed, despite the potential for pseudoprimes. So, we examine the commonly used Miller-Rabin and Lucas tests, showing that numbers with many nonwitnesses are usually Carmichael or Lucas-Carmichael numbers in a specific form. We then use these categorizations, through a generalization of Korselt’s criterion, to prove that there are no numbers with many nonwitnesses for both tests, affirming the two tests’ relative independence. As Carmichael and Lucas-Carmichael numbers are in general more difficult for the two tests to deal with, we next search for numbers which are both Carmichael and Lucas-Carmichael numbers, experimentally finding none less than 1016. We thus conjecture that there are no such composites and, using multi- variate calculus with symmetric polynomials, begin developing techniques to prove this. 2 1 Introduction In the current information age, cryptographic systems to protect data have become a funda- mental necessity. With the quantity of data distributed over the internet, the importance of encryption for protecting individual privacy has greatly risen. Indeed, according to [EMC16], cryptography is allows for authentication and protection in online commerce, even when working with vital financial information (e.g. in online banking and shopping). Now that more and more transactions are done through the internet, rather than in person, the importance of secure encryption schemes is only growing.
    [Show full text]
  • Idempotent Integers: the Complete Class of Numbers N = ¯P¯Q That Work
    Idempotent Integers: The complete class of numbers n =p ¯q¯ that work correctly in RSA A Mathematical Curiosity Barry Fagin Senior Associate Dean of the Faculty Professor of Computer Science United States Air Force Academy 2021 Conference on Geometry, Algebraic Number Theory and Applications Barry Fagin (A Mathematical Curiosity) Take two positive numbers p¯; q¯. Compute n =p ¯q¯. Pick e; d such that ed ≡ 1. (¯p−1)(¯q−1) What are the conditions on p¯; q¯ such that s.t. 1 ed ? 8a 2 Zn; 8e; d ed ≡ ; a ≡ a (¯p−1)(¯q−1) n In other words, what are the conditions on p¯; q¯ such that RSA works correctly? Note: not asking about security. Barry Fagin (A Mathematical Curiosity) 70's: p¯; q¯ prime meets the conditions. If suciently large, system is believed to be secure due to computational intractability of factoring. 90's: Some Carmichael numbers C also can be factored into p¯; q¯ that meet the required conditions. This work (since 2018) gives necessary and sucient conditions, shows examples. Barry Fagin (A Mathematical Curiosity) Cut to the chase Answer is: p¯; q¯ square free, gcd(p¯; q¯) = 1 such that (¯p − 1)(¯q − 1) ≡ 0 λ(n) where λ denotes the Carmichael lambda function. Barry Fagin (A Mathematical Curiosity) From n's point of view Equivalently, a square-free n can be factored into n =p ¯q¯ such that (¯p − 1)(¯q − 1) ≡ 0 λ(n) In that case, we say that n has an idempotent factorization, and that n is an idempotent integer. Barry Fagin (A Mathematical Curiosity) The rst 8 square-free n with m ≥ 3 factors that admit idempotent factorizations are shown below: n p or p¯ q¯ 30 5 6 42 7 6 66 11 6 78 13 6 102 17 6 105 7 15 114 19 6 130 13 10 Table: Values of n that admit idempotent factorizations Barry Fagin (A Mathematical Curiosity) Idempotent factorizations where one of p¯; q¯ is prime and one is composite are semi-composite.
    [Show full text]
  • Fermat Pseudoprimes
    1 TWO HUNDRED CONJECTURES AND ONE HUNDRED AND FIFTY OPEN PROBLEMS ON FERMAT PSEUDOPRIMES (COLLECTED PAPERS) Education Publishing 2013 Copyright 2013 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599732572 ISBN: 978-1-59973-257-2 1 INTRODUCTION Prime numbers have always fascinated mankind. For mathematicians, they are a kind of “black sheep” of the family of integers by their constant refusal to let themselves to be disciplined, ordered and understood. However, we have at hand a powerful tool, insufficiently investigated yet, which can help us in understanding them: Fermat pseudoprimes. It was a night of Easter, many years ago, when I rediscovered Fermat’s "little" theorem. Excited, I found the first few Fermat absolute pseudoprimes (561, 1105, 1729, 2465, 2821, 6601, 8911…) before I found out that these numbers are already known. Since then, the passion for study these numbers constantly accompanied me. Exceptions to the above mentioned theorem, Fermat pseudoprimes seem to be more malleable than prime numbers, more willing to let themselves to be ordered than them, and their depth study will shed light on many properties of the primes, because it seems natural to look for the rule studying it’s exceptions, as a virologist search for a cure for a virus studying the organisms that have immunity to the virus.
    [Show full text]
  • Improving the Speed and Accuracy of the Miller-Rabin Primality Test
    Improving the Speed and Accuracy of the Miller-Rabin Primality Test Shyam Narayanan Mentor: David Corwin MIT PRIMES-USA Abstract Currently, even the fastest deterministic primality tests run slowly, with the Agrawal- Kayal-Saxena (AKS) Primality Test runtime O~(log6(n)), and probabilistic primality tests such as the Fermat and Miller-Rabin Primality Tests are still prone to false re- sults. In this paper, we discuss the accuracy of the Miller-Rabin Primality Test and the number of nonwitnesses for a composite odd integer n. We also extend the Miller- '(n) Rabin Theorem by determining when the number of nonwitnesses N(n) equals 4 5 and by proving that for all n, if N(n) > · '(n) then n must be of one of these 3 32 forms: n = (2x + 1)(4x + 1), where x is an integer, n = (2x + 1)(6x + 1), where x is an integer, n is a Carmichael number of the form pqr, where p, q, r are distinct primes congruent to 3 (mod 4). We then find witnesses to certain forms of composite numbers with high rates of nonwitnesses and find that Jacobi nonresidues and 2 are both valuable bases for the Miller-Rabin test. Finally, we investigate the frequency of strong pseudoprimes further and analyze common patterns using MATLAB. This work is expected to result in a faster and better primality test for large integers. 1 1 Introduction Data is growing at an astoundingly rapid rate, and better information security is re- quired to protect increasing quantities of data. Improved data protection requires more sophisticated cryptographic methods.
    [Show full text]
  • On $ K $-Lehmer Numbers
    ON k-LEHMER NUMBERS JOSE´ MAR´IA GRAU AND ANTONIO M. OLLER-MARCEN´ Abstract. Lehmer’s totient problem consists of determining the set of pos- itive integers n such that ϕ(n)|n − 1 where ϕ is Euler’s totient function. In this paper we introduce the concept of k-Lehmer number. A k-Lehmer num- ber is a composite number such that ϕ(n)|(n − 1)k. The relation between k-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also k-Lehmer numbers. AMS 2000 Mathematics Subject Classification: 11A25,11B99 1. Introduction Lehmer’s totient problem asks about the existence of a composite number such that ϕ(n)|(n − 1), where ϕ is Euler’s totient function. Some authors denote these numbers by Lehmer numbers. In 1932, Lehmer (see [13]) showed that every Lemher numbers n must be odd and square-free, and that the number of distinct prime factors of n, d(n), must satisfy d(n) > 6. This bound was subsequently extended to d(n) > 10. The current best result, due to Cohen and Hagis (see [9]), is that n must have at least 14 prime factors and the biggest lower bound obtained for such numbers is 1030 (see [17]). It is known that there are no Lehmer numbers in certain sets, such as the Fibonacci sequence (see [15]), the sequence of repunits in base g for any g ∈ [2, 1000] (see [8]) or the Cullen numbers (see [11]).
    [Show full text]
  • 1 Fermat Pseudoprimes 2 Carmichael Numbers
    Z∗ Z∗ MATH/CSCI 4116: CRYPTOGRAPHY, FALL 2014 Then G is a subgroup of n, thus |G| 6 | n|. Since n is not a Fermat pseudo- Z∗ Z∗ Handout 3: The Miller-Rabin Primality Test prime, there exists some b ∈ n with b 6∈ G, thus |G| < | n|. It follows that 1 Z∗ n−1 |G| 6 2 | n| 6 2 . Finally, whenever b ∈ {1,...,n − 1} and b 6∈ G, then b Peter Selinger n−1 fails the test; there are at least 2 such elements. Algorithm 1.3 (Fermat pseudoprime test). 1 Fermat Pseudoprimes Input: Integers n > 2 and t > 1. Output: If n is prime, output “yes”. If n is not a Fermat pseudoprime, output “no” A primality test is an algorithm that, given an integer n, decides whether n is prime with probability at least 1 − 1/2t, “yes” with probability at most 1/2t. or not. The most naive algorithm, trial division, is hopelessly inefficient when n is Algorithm: Pick t independent, uniformly distributed random numbers b1,...,bt ∈ very large. Fortunately, there exist much more efficient algorithms for determining −1 . If n mod for all , output “yes”, else output “no”. whether n is prime. The most common such algorithms are probabilistic; they {1,...,n − 1} bi ≡ 1( n) i give the correct answer with very high probability. All efficient primality testing algorithms are based, in one way or another, on Fermat’s Little Theorem. Proof. We prove that the output of the algorithm is as specified. If n is prime, then the algorithm outputs “yes” by Fermat’s Little Theorem.
    [Show full text]
  • [Math.NT] 17 Apr 2006 Esthan Less Tms 1Piefcos Elet We Factors
    THE CARMICHAEL NUMBERS UP TO 1018 RICHARD G.E. PINCH Abstract. We extend our previous computations to show that there are 1401644 Carmichael numbers up to 1018. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a “large prime variation”. We present further statistics on the distribution of Carmichael numbers. 1. Introduction A Carmichael number N is a composite number N with the property that for every b prime to N we have bN−1 ≡ 1 mod N. It follows that a Carmichael number N must be square-free, with at least three prime factors, and that p − 1|N − 1 for every prime p dividing N: conversely, any such N must be a Carmichael number. For background on Carmichael numbers and details of previous computations we refer to our previous paper [4]: in that paper we described the computation of the Carmichael numbers up to 1015 and presented some statistics. These computa- tions have since been extended to 1016 [5], 1017 [6] and now to 1018, using similar techniques, and we present further statistics. The complete list of Carmichael numbers up to 1018 is available from the author’s web site at www.chalcedon.demon.co.uk. 2. Organisation of the search We used improved versions of strategies first described in [4]. The principal search was a depth-first back-tracking search over possible se- r d quences of primes factors p1,...,pd. Put Pr = i=1 pi, Qr = i=r+1 pi and Lr = lcm {pi − 1 : i =1,...,r}.
    [Show full text]
  • Mathematics 4: Number Theory Problem Sheet 3 Workshop 26 Oct
    Mathematics 4: Number Theory Problem Sheet 3 Workshop 26 Oct 2012 The aim of this workshop is to show that Carmichael numbers are squarefree and have at least 3 distinct prime factors. (1) (Warm-up question.) Show that n > 1 is prime iff an−1 ≡ 1 (mod n) for 1 ≤ a ≤ n − 1. If n is prime, then the result is true by Fermat’s Theorem. If n is composite, then, for a = p, a prime factor of n, the equation pn−1 + kn =1 has no solution, since the LHS is divisible by p. So result is true iff n prime. Recall that a positive integer is said to be squarefree if it is not divisible by the square of any prime number. Recall too that a Carmichael number is a composite number n with the property that for every integer a coprime to n we have an−1 ≡ 1 (mod n). (2) Proving that Carmichael numbers are squarefree. (a) Show that a given nonsquarefree number n can be written in the form n = pℓN for some prime p and integers N and ℓ with ℓ ≥ 2 and gcd(p, N)=1. (b) Show that (1 + pN)n−1 6≡ 1 (mod p2). (c) Deduce that Carmichael numbers are squarefree. (a) Take a prime p such that p2 | n. Then pℓ||n for some ℓ ≥ 2. So n = pℓN say, where N = n/pℓ is coprime to p. (b) Now (1+pN)n−1 ≡ 1+ pN(n − 1) (mod p2) ≡ 1 − pN (mod p2) 6≡ 1 (mod p2), by the Binomial Theorem and because p2 | n and gcd(N,p)=1.
    [Show full text]